# Measures of Center

Sample mean versus population mean

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## Key Questions

• The common measures of center or measures of central tendency are the mean, median and mode

For the mean we have the common average called arithmetic mean simply sum of all values divided by the total number of values. We have the population mean denoted by $\mu$ and sample mean denoted by $\overline{x}$ the definitional formulas are

$\mu = \frac{\sum x}{N} \mathmr{and} \overline{x} = \frac{\sum x}{n}$

While median is the middle observation of an arrayed data. Meaning median is a positional measure. That is denoted by ${\mu}_{d} \mathmr{and} M d$

if the number of observation is odd:

${\mu}_{d} = {x}_{\frac{n + 1}{2}}$ and if the number of observation is even

${\mu}_{d} = \frac{{x}_{\frac{n}{2}} + {x}_{\frac{n}{2} + 1}}{2}$ for illustrative example find the median of the given scores: 2, 7, 8, 1, 4, 5

First thing to do is make an array. That is to arrange the data from least-to-greatest or greatest-to-least.

Array: 1, 2, 4, 5, 7, 8
Next step: since the number of observation is even (6) use the appropriate formula that is
${\mu}_{d} = \frac{{x}_{\frac{n}{2}} + {x}_{\frac{n}{2} + 1}}{2} = \frac{{x}_{\frac{6}{2}} + {x}_{\frac{6}{2} + 1}}{2} = \frac{{x}_{3} + {x}_{4}}{2}$
it tells that you need to add the 3rd and 4th observations and divide it by two.

Now we have

${\mu}_{d} = \frac{4 + 5}{2} = 4.5$ Therefore the median score is 4.5.

Lastly, mode is the most observed value or values. That is denoted by ${\mu}_{o} \mathmr{and} M o$. For example,

Find the modal score of the given set: 1, 1, 2, 2, 4, 5, 5, 5

5 is the most observed value therefore Mo = 5. Another example find the modal score of the set: 1, 1, 1, 2, 2, 2, 3, 3,3

Since each observation (1 , 2, 3) appears thrice, there is no mode. There is no most observed value.

Note if a set has exactly one mode then the set is unimodal, if there is two modes say for the set: 1, 1, 2, 3, 3 the modes are 1 and 3 then, this set is bimodal. A set can also be trimodal or multimodal.

Supplementary: Geometric mean, harmonic mean. Hope this help.

• For the mean we have the common average called arithmetic mean simply sum of all values divided by the total number of values. We have the population mean denoted by $\mu$ and sample mean denoted by $\overline{x}$ the definitional formulas are

$\mu = \frac{\sum x}{N} \mathmr{and} \overline{x} = \frac{\sum x}{n}$

While median is the middle observation of an arrayed data. Meaning median is a positional measure. That is denoted by ${\mu}_{d} \mathmr{and} M d$

if the number of observation is odd:

${\mu}_{d} = {x}_{\frac{n + 1}{2}}$ and if the number of observation is even

${\mu}_{d} = \frac{{x}_{\frac{n}{2}} + {x}_{\frac{n}{2} + 1}}{2}$ for illustrative example find the median of the given scores: 2, 7, 8, 1, 4, 5

First thing to do is make an array. That is to arrange the data from least-to-greatest or greatest-to-least.

Array: 1, 2, 4, 5, 7, 8
Next step: since the number of observation is even (6) use the appropriate formula that is
${\mu}_{d} = \frac{{x}_{\frac{n}{2}} + {x}_{\frac{n}{2} + 1}}{2} = \frac{{x}_{\frac{6}{2}} + {x}_{\frac{6}{2} + 1}}{2} = \frac{{x}_{3} + {x}_{4}}{2}$
it tells that you need to add the 3rd and 4th observations and divide it by two.

Now we have

${\mu}_{d} = \frac{4 + 5}{2} = 4.5$ Therefore the median score is 4.5.

• When a dataset has a few very extreme cases.

Example: We have a dataset of 1000 in which most values hover around the 1000-mark. Let's say the mean and the median are both 1000. Now we add one 'millionaire'. The mean will rise dramatically to almost 2000, while the median will not really change, because it will be the value of case 501 in stead of the in-between of case 500 and case 501 (cases arranged in order of value)

• Population mean is the true average of interest which is usually known or can be obtained by incorporating the whole population,
while the sample mean is calculated from a sample which is a representative part of the population.

Sample mean may not be exactly equal to the true average but probably very close to the true value of the population mean.

Population mean is the parameter and sample mean is the estimator of the parameter when the true mean is unknown

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